## Unified perfectly matched layer for finite-difference time-domain modeling of dispersive optical materials

Optics Express, Vol. 17, Issue 23, pp. 21179-21190 (2009)

http://dx.doi.org/10.1364/OE.17.021179

Acrobat PDF (272 KB)

### Abstract

Finite-difference time-domain (FDTD) simulations of any electromagnetic problem require truncation of an often-unbounded physical region by an electromagnetically bounded region by deploying an artificial construct known as the perfectly matched layer (PML). As it is not possible to construct a universal PML that is non-reflective for different materials, PMLs that are tailored to a specific problem are required. For example, depending on the number of dispersive materials being truncated at the boundaries of a simulation region, an FDTD code may contain multiple sets of update equations for PML implementations. However, such an approach is prone to introducing coding errors. It also makes it extremely difficult to maintain and upgrade an existing FDTD code. In this paper, we solve this problem by developing a new, unified PML algorithm that can effectively truncate all types of linearly dispersive materials. The unification of the algorithm is achieved by employing a general form of the medium permittivity that includes three types of dielectric response functions, known as the Debye, Lorentz, and Drude response functions, as particular cases. We demonstrate the versatility and flexibility of the new formulation by implementing a single FDTD code to simulate absorption of electromagnetic pulse inside a medium that is adjacent to dispersive materials described by different dispersion models. The proposed algorithm can also be used for simulations of optical phenomena in metamaterials and materials exhibiting negative refractive indices.

© 2009 Optical Society of America

## 1. Introduction

1. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**, 302–307 (
1966). [CrossRef]

3. P. Petropoulos, “Stability and phase error analysis of FD-TD in dispersive dielectrics,” IEEE Trans. Antennas Propag. **42**, 62–69 (
1994). [CrossRef]

4. M. Premaratne and S. K. Halgamuge, “Rigorous analysis of numerical phase and group velocity bounds in Yee’s FDTD grid,” IEEE Microwave Wirel. Compon. Lett. **17**, 556–558 (
2007). [CrossRef]

5. R. Luebbers, F. Hunsberger, K. Kunz, R. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. **32**, 222–227 (
1990). [CrossRef]

6. D. Kelley and R. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propag. **44**, 792–797 (
1996). [CrossRef]

7. J. Young, “Propagation in linear dispersive media: Finite difference time-domain methodologies,” IEEE Trans. Antennas Propag. **43**, 422–426 (
1995). [CrossRef]

8. D. Sullivan, “Z-transform theory and the FDTD method,” IEEE Trans. Antennas Propag. **44**, 28–34 (
1996). [CrossRef]

9. M. Okoniewski, M. Mrozowski, and M. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microwave Guided Wave Lett. **7**, 121–123 (
1997). [CrossRef]

10. Y. Takayama and W. Klaus, “Reinterpretation of the auxiliary differential equation method for FDTD,” IEEE Microwave Wirel. Compon. Lett. **12**, 102–104 (
2002). [CrossRef]

*et al*. showed that dispersive materials described by the Debye and Lorentz models can be represented by a generic expression template consisting of complex-conjugate pairs of simple residues [11

11. M. Han, R. Dutton, and S. Fan, “Model dispersive media in finite-difference time-domain method with complex-conjugate pole-residue pairs,” IEEE Microwave Compon. Lett. **16**, 119–121 (
2006). [CrossRef]

12. K. P. Prokopidis, “On the development of efficient FDTD-PML formulations for general dispersive media,” Int. J. Numer. Model. **21**, 395–411 (
2008). [CrossRef]

## 2. Construction of unified PML for arbitrary dispersive media

*µ*, throughout this study. According to Han

*et al*. [11

11. M. Han, R. Dutton, and S. Fan, “Model dispersive media in finite-difference time-domain method with complex-conjugate pole-residue pairs,” IEEE Microwave Compon. Lett. **16**, 119–121 (
2006). [CrossRef]

*a*and

_{p}*c*as follows:

_{p}*Δ*

^{a}*ε*is the weight of the

_{p}*p*th pole and

*τ*is the carrier relaxation time.

_{p}

^{b}*ω*is the resonance frequency and

_{p}*δ*is the damping parameter; parameters

_{p}*a*

_{0}and

*c*

_{0}are the same as for Debye model.

^{c}*ω*

_{pl}and

*ω*are the plasma and collision frequencies, respectively.

_{c}*ε*

_{∞}is the real-valued, high-frequency permittivity,

*P*is a positive integer, and an asterisk denotes complex conjugation. Hereafter, we use a tilde above a time-domain function to denote its Fourier transform in the frequency domain, i.e.,

*a*, and residues,

_{p}*c*, entering Eq. (1) depend on a particular model of dispersive media. For generality of this treatment, we refer to materials that are described by Eq. (1) as complex-conjugate-pole residue (CCPR) medium from here onwards. Table 1 provides expressions of permittivities and summarizes values of the parameters

_{p}*a*and

_{p}*c*for three widely used models of dielectric response from dispersive media [2, 13

_{p}13. R. Luebbers, F. Hunsberger, and K. Kunz, “A frequency-dependent finite-difference time-domain formulation for transient propagation in plasma,” IEEE Trans. Antennas Propag. **39**, 29–34 (
1991). [CrossRef]

_{0}=-

*iε*

_{0}lim

_{ω→0}

*ω*ε

*̃*(

*ω*) into the expressions of dielectric permittivities. The Drude model does not require this modification as it accounts for σ

_{0}=ε

_{0}

*ω*

^{2}

_{pl}

*ω*

^{-1}

_{c}automatically.

## 2.1. Synthesis of PML for CCPR media

**E**̃(

**r**,

*ω*) and

**H**̃(

**r**,

*ω*) be the electric and magnetic fields inside a CCPR medium. In the absence of external sources, the evolution of these fields (in the frequency domain) is governed by the Maxwell equations,

*µ*

_{0}and

*ε*

_{0}are the magnetic and dielectric permittivities of free space. An absorbing PML that is perfectly impedance-matched to its adjacent media, can be created by introducing a specific transformation of the Cartesian coordinates (for details see [14

14. W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Tech. Lett **7**, 599–604 (
1994). [CrossRef]

15. S. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. **44**, 1630–1639 (
1996). [CrossRef]

*ε*̃ (

*ω*)

*µ*

*s*(

_{j}*j*=

*x*,

*y*,

*z*), the impedances of CCPR medium and PML are equal with the value

*s*appropriately. In the present paper, we use the following definition [16

_{j}16. M. Kuzuoglu and R. Mittra, “Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers,” IEEE Microwave Guided Wave Lett. **6**, 447–449 (
1996). [CrossRef]

19. J.-P. Berenger, “Numerical reflection from FDTD-PMLs: A comparison of the split PML with the unsplit and CFS PMLs,” IEEE Trans. Antennas Propag. **50**, 258–265 (
2002). [CrossRef]

18. D. Correia and J.-M. Jin, “Performance of regular PML, CFS-PML, and second-order PML for waveguide problems,” Microwave Opt. Technol. Lett. **48**, 2121–2126 (
2006). [CrossRef]

*>0,*

_{j}*κ*>1, and

_{j}*γ*>0 on the attenuation properties of CFS PML is thoroughly analyzed in Refs. [19

19. J.-P. Berenger, “Numerical reflection from FDTD-PMLs: A comparison of the split PML with the unsplit and CFS PMLs,” IEEE Trans. Antennas Propag. **50**, 258–265 (
2002). [CrossRef]

21. J.-P. Berenger, “Application of the CFS PML to the absorption of evanescent waves in waveguides,” IEEE Microwave Wirel. Compon. Lett. **12**, 218–220 (
2002). [CrossRef]

20. D. Correia and J.-M. Jin, “On the development of a higher-order PML,” IEEE Trans. Antennas Propag. **53**, 4157–4163 (
2005). [CrossRef]

*resembles the conductivity profile of PML and provides attenuation of propagating waves in the*

_{j}*j*th direction, whereas

*κ*and

_{j}*γ*absorb evanescent waves. To avoid numerical reflections from the PML boundaries, the parameters σ

*and*

_{j}*κ*need to be smoothly varied in space. One choice of these parameters and the guidelines for the coefficients involved can be found in [22

_{j}22. Y. Rickard and N. Georgieva, “Problem-independent enhancement of PML ABC for the FDTD method,” IEEE Trans. Antennas Propag. **51**, 3002–3006(
2003). [CrossRef]

*k*≤

*δ*,

_{j}*δ*is the PML depth in the

_{j}*j*th direction,

*m*∈ [-3,3] and

*n*∈ [2,6] are the user-defined integers meeting the condition

*m*+

*n*> 1,

*κ*

_{max}∈ (1,10], σ

_{max}=-

*cε*

_{0}(

*m*+

*n*+1) ln

*R*

_{0}/(2

*δ*),

_{j}*c*is the speed of light in vacuum, and

*R*

_{0}∈ [10

^{-12},10

^{-2}].

## 2.2. Derivation of FDTD update equations

**E**(

**r**,

*t*) and

**H**(

**r**,

*t*).

**R**

*and*

_{E}**R**

*can be derived as follows. Substituting Eq. (1) into Eq. (3b) and omitting the dash sign for convenience, we are led to the equation*

_{H}**R**

*:*

_{E}## 2.3. Restoration of the electromagnetic field inside PML

*iω*↔

*∂*/

*∂t*and discretizing the resulting differential equations with appropriate difference operators. This procedure, however, is quite tedious in our case and becomes even harder for more complicated forms of the coefficients

*s*(e.g., for negative-index-materials [23

_{j}23. S. Cummer, “Perfectly matched layer behavior in negative refractive index materials,” IEEE Antennas Wirel. Propag. Lett. **3**, 172–175 (
2004). [CrossRef]

20. D. Correia and J.-M. Jin, “On the development of a higher-order PML,” IEEE Trans. Antennas Propag. **53**, 4157–4163 (
2005). [CrossRef]

8. D. Sullivan, “Z-transform theory and the FDTD method,” IEEE Trans. Antennas Propag. **44**, 28–34 (
1996). [CrossRef]

*x*component of the electric field using the definition,

*R*̃

_{E,x}=

_{x}*E*̃

*. Introducing new parameters,*

_{x}*ξ*=

_{j}*ξ*

_{0}+σ

*/(*

_{j}*κ*

_{j}*ε*

_{0}),

*ξ*

_{0}=

*γ*/

*ε*

_{0}, and utilizing Eq. (4), the relation between

*R*̃

_{E,x}and

*E*̃

*takes the form*

_{x}*z*

^{-k}

*A*(

^{n}*z*)↔

*A*

^{n-k}, we get

*E*and

_{y}*E*components from Eq. (11) upon index permutations

_{z}*x*⇄

*y*and

*x*⇄

*z*, respectively. The components

*H*

_{n+1j}satisfy equations identical to those for

*E*

^{n+1}

*if we use*

_{j}*R*

^{k}_{H,j}in place of

*R*

^{k}_{E,j}.

## 2.4. Steps to update FDTD grid

**R**

^{n+1/2}

*values. The back-stored values of*

_{E}**H**

*,*

^{n}**J**

^{n-1/2}

*, and*

_{p}**R**

^{n-1/2}

*are required for this update.*

_{E}**J**

^{n+1/2}

*for each pole*

_{p}*p*using Eq. (8). The value of

**R**

^{n+1/2}

*are then calculated for the current time step using the back-stored values of*

_{E}**J**

^{n-1/2}

*and*

_{p}**R**

^{n-1/2}

*.*

_{E}*y*and

*z*components to restore the electric field

**E**

^{n+1/2}from the back-stored values of

**E**

^{n-1/2},

**E**

^{n-3/2},

**R**

^{n-1/2}

*, and*

_{E}**R**

^{n-3/2}

*as well as from*

_{E}**R**

^{n+1/2}

*calculated during the current time step.*

_{E}**R**

^{n+1}

*values using Eq. (10). The back-stored values of*

_{H}**R**

*and*

^{n}_{H}**E**

^{n+1/2}are required for this update.

**H**

^{n+1}using back-stored values of

**H**

*,*

^{n}**H**

^{n-1},

**R**

*, and*

^{n}_{H}**R**

^{n-1}

*as well as the value of*

_{H}**R**

^{n+1}

*calculated during the current time step.*

_{H}**R**

*and*

^{n}_{E}**R**

*should be replaced by*

^{n}_{H}**E**

*and*

^{n}**H**

*in the other steps. We used these steps to create an FDTD code for numerical simulations presented in the next section.*

^{n}## 3. Numerical examples

*E*, calculated using a 50×50-cell grid truncated with a PML, from that calculated on a reference grid with 400×400 cells. Mathematically, this deviation may be characterized by the global error [12

_{z}12. K. P. Prokopidis, “On the development of efficient FDTD-PML formulations for general dispersive media,” Int. J. Numer. Model. **21**, 395–411 (
2008). [CrossRef]

25. J.-P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **127**, 363–379 (
1996). [CrossRef]

*n*is the time-step number and

*f*is the carrier frequency.

_{c}*x*and

*y*directions and its parameters (see page 5) are empirically chosen to be

*m*=0,

*n*=4,

*κ*

_{max}=2,

*γ*=1, and

*R*

_{0}=10

^{-7}. The cell dimensions Δ

*x*=Δ

*y*=

*λ*

_{min}/40, where

*λ*

_{min}=

*c*/(2.5

*f*) is the wavelength of interest, and the Courant factor

_{c}*S*=

*c*Δ

*t*/Δ

*x*=0.1.

26. O. P. Gandhi, B.-Q. Gao, and J.-Y. Chen, “A frequency-dependent finite-difference time-domain formulation for general dispersive media,” IEEE Trans. Microwave Theory Tech. **41**, 658–665 (
1993). [CrossRef]

27. W. D. Hurt, “Multiterm Debye dispersion relations for permittivity of muscle,” IEEE Trans. Bio. Eng. **32**, 60–64 (
1985). [CrossRef]

28. K. E. Oughstun, “Computational methods in ultrafast time-domain optics,” Comput. Sci. Eng. **5**, 22–32 (
2003). [CrossRef]

12. K. P. Prokopidis, “On the development of efficient FDTD-PML formulations for general dispersive media,” Int. J. Numer. Model. **21**, 395–411 (
2008). [CrossRef]

29. J. Xi and M. Premaratne, “Analysis of the optical force dependency on beam polarization: Dielectric/metallic spherical shell in a Gaussian beam,” J. Opt. Soc. Am. B **26**, 973–980 (
2009). [CrossRef]

30. A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B **71**, 085416(1–7) (
2005). [CrossRef]

*a*and

_{p}*c*using Tables 1 and 2. When only the experimental spectrum of dielectric response function is known, one needs to employ curve fitting using Eq. (1). In the simulations, we chose the carrier frequency of the excitation pulse to be 1 GHz for the Debye and Drude media and 500 THz for the Lorentz and Drude–Lorentz media. Figure 2 shows evolution of global error with time for sample media truncated by PMLs of different thicknesses. As seen there, the global error is relatively small in the case of an 8-cell PML (blue curves) for all four dispersive media. In particular, the maximum value of

_{p}*χ*

^{2}is at most ~10

^{-3}, and can be ~10

^{-6}for the Drude model, signifying high efficiency of the PML (for comparison, see Ref. [12

**21**, 395–411 (
2008). [CrossRef]

*f*=500 THz is launched at the center of 380×380-cell grid surrounded by a 10-cell PML.

_{c}*n*=2300 (left panel) and

*n*=3300 (right panel). One can see that reflections from all parts of the PML are almost negligible for both the oblique and grazing waves. It should also be noted that the PML provides efficient absorption for both the high-frequency (left panel) and low-frequency (right panel) field components. These conclusions are supported by the evolution of global error shown by the red curve in Fig. 4. Particularly,

*χ*

^{2}≈3×10

^{-8}for the left panel in Fig. 3, but it becomes ≈1.2×10

^{-3}for the right panel in Fig. 3. As expected, the global error drastically depends on the PML thickness, increasing by more than a factor of 20 for a 5-cell PML (green curve in Fig. 4) and decreasing nearly by a factor of 6 for a 15-cell PML (blue curve in Fig. 4) at

*n*=3300.

## 4. Conclusion

## Acknowledgments

## References and links

1. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

2. | A. Taflove and S. C. Hagness, |

3. | P. Petropoulos, “Stability and phase error analysis of FD-TD in dispersive dielectrics,” IEEE Trans. Antennas Propag. |

4. | M. Premaratne and S. K. Halgamuge, “Rigorous analysis of numerical phase and group velocity bounds in Yee’s FDTD grid,” IEEE Microwave Wirel. Compon. Lett. |

5. | R. Luebbers, F. Hunsberger, K. Kunz, R. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. |

6. | D. Kelley and R. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propag. |

7. | J. Young, “Propagation in linear dispersive media: Finite difference time-domain methodologies,” IEEE Trans. Antennas Propag. |

8. | D. Sullivan, “Z-transform theory and the FDTD method,” IEEE Trans. Antennas Propag. |

9. | M. Okoniewski, M. Mrozowski, and M. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microwave Guided Wave Lett. |

10. | Y. Takayama and W. Klaus, “Reinterpretation of the auxiliary differential equation method for FDTD,” IEEE Microwave Wirel. Compon. Lett. |

11. | M. Han, R. Dutton, and S. Fan, “Model dispersive media in finite-difference time-domain method with complex-conjugate pole-residue pairs,” IEEE Microwave Compon. Lett. |

12. | K. P. Prokopidis, “On the development of efficient FDTD-PML formulations for general dispersive media,” Int. J. Numer. Model. |

13. | R. Luebbers, F. Hunsberger, and K. Kunz, “A frequency-dependent finite-difference time-domain formulation for transient propagation in plasma,” IEEE Trans. Antennas Propag. |

14. | W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Tech. Lett |

15. | S. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. |

16. | M. Kuzuoglu and R. Mittra, “Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers,” IEEE Microwave Guided Wave Lett. |

17. | S. Gedney, “Scaled CFS-PML: It is more robust, more accurate, more efficient, and simple to implement. Why aren’t you using it?” in |

18. | D. Correia and J.-M. Jin, “Performance of regular PML, CFS-PML, and second-order PML for waveguide problems,” Microwave Opt. Technol. Lett. |

19. | J.-P. Berenger, “Numerical reflection from FDTD-PMLs: A comparison of the split PML with the unsplit and CFS PMLs,” IEEE Trans. Antennas Propag. |

20. | D. Correia and J.-M. Jin, “On the development of a higher-order PML,” IEEE Trans. Antennas Propag. |

21. | J.-P. Berenger, “Application of the CFS PML to the absorption of evanescent waves in waveguides,” IEEE Microwave Wirel. Compon. Lett. |

22. | Y. Rickard and N. Georgieva, “Problem-independent enhancement of PML ABC for the FDTD method,” IEEE Trans. Antennas Propag. |

23. | S. Cummer, “Perfectly matched layer behavior in negative refractive index materials,” IEEE Antennas Wirel. Propag. Lett. |

24. | A. V. Oppenheim and R. W. Schafer, |

25. | J.-P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

26. | O. P. Gandhi, B.-Q. Gao, and J.-Y. Chen, “A frequency-dependent finite-difference time-domain formulation for general dispersive media,” IEEE Trans. Microwave Theory Tech. |

27. | W. D. Hurt, “Multiterm Debye dispersion relations for permittivity of muscle,” IEEE Trans. Bio. Eng. |

28. | K. E. Oughstun, “Computational methods in ultrafast time-domain optics,” Comput. Sci. Eng. |

29. | J. Xi and M. Premaratne, “Analysis of the optical force dependency on beam polarization: Dielectric/metallic spherical shell in a Gaussian beam,” J. Opt. Soc. Am. B |

30. | A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(160.4670) Materials : Optical materials

(260.2030) Physical optics : Dispersion

(080.1753) Geometric optics : Computation methods

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 23, 2009

Revised Manuscript: October 23, 2009

Manuscript Accepted: November 4, 2009

Published: November 6, 2009

**Citation**

Indika Udagedara, Malin Premaratne, Ivan D. Rukhlenko, Haroldo T. Hattori, and Govind P. Agrawal, "Unified perfectly matched layer for finite-difference time-domain modeling
of dispersive optical materials," Opt. Express **17**, 21179-21190 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-23-21179

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### References

- K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
- P. Petropoulos, "Stability and phase error analysis of FD-TD in dispersive dielectrics," IEEE Trans. Antennas Propag. 42, 62-69 (1994). [CrossRef]
- M. Premaratne and S. K. Halgamuge, "Rigorous analysis of numerical phase and group velocity bounds in Yee’s FDTD grid," IEEE Microwave Wirel. Compon. Lett. 17, 556-558 (2007). [CrossRef]
- R. Luebbers, F. Hunsberger, K. Kunz, R. Standler, and M. Schneider, "A frequency-dependent finite-difference time-domain formulation for dispersive materials," IEEE Trans. Electromagn. Compat. 32, 222-227 (1990). [CrossRef]
- D. Kelley and R. Luebbers, "Piecewise linear recursive convolution for dispersive media using FDTD," IEEE Trans. Antennas Propag. 44, 792-797 (1996). [CrossRef]
- J. Young, "Propagation in linear dispersive media: Finite difference time-domain methodologies," IEEE Trans. Antennas Propag. 43, 422-426 (1995). [CrossRef]
- D. Sullivan, "Z-transform theory and the FDTD method," IEEE Trans. Antennas Propag. 44, 28-34 (1996). [CrossRef]
- M. Okoniewski, M. Mrozowski, and M. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Microwave Guided Wave Lett. 7, 121-123 (1997). [CrossRef]
- Y. Takayama and W. Klaus, "Reinterpretation of the auxiliary differential equation method for FDTD," IEEE Microwave Wirel. Compon. Lett. 12, 102-104 (2002). [CrossRef]
- M. Han, R. Dutton, and S. Fan, "Model dispersive media in finite-difference time-domain method with complexconjugate pole-residue pairs," IEEE Microwave Compon. Lett. 16, 119-121 (2006). [CrossRef]
- K. P. Prokopidis, "On the development of efficient FDTD-PML formulations for general dispersive media," Int. J. Numer. Model. 21, 395-411 (2008). [CrossRef]
- R. Luebbers, F. Hunsberger, and K. Kunz, "A frequency-dependent finite-difference time-domain formulation for transient propagation in plasma," IEEE Trans. Antennas Propag. 39, 29-34 (1991). [CrossRef]
- W. C. Chew and W. H. Weedon, "A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates," Microwave Opt. Tech. Lett 7, 599-604 (1994). [CrossRef]
- S. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propag. 44, 1630-1639 (1996). [CrossRef]
- M. Kuzuoglu and R. Mittra, "Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers," IEEE Microwave Guided Wave Lett. 6, 447-449 (1996). [CrossRef]
- S. Gedney, "Scaled CFS-PML: It is more robust, more accurate, more efficient, and simple to implement. Why aren’t you using it?" in Antennas and Propagation Society International Symposium 364-367 (2005).
- D. Correia and J.-M. Jin, "Performance of regular PML, CFS-PML, and second-order PML for waveguide problems," Microwave Opt. Technol. Lett. 48, 2121-2126 (2006). [CrossRef]
- J.-P. Berenger, "Numerical reflection from FDTD-PMLs: A comparison of the split PML with the unsplit and CFS PMLs," IEEE Trans. Antennas Propag. 50, 258-265 (2002). [CrossRef]
- D. Correia and J.-M. Jin, "On the development of a higher-order PML," IEEE Trans. Antennas Propag. 53, 4157-4163 (2005). [CrossRef]
- J.-P. Berenger, "Application of the CFS PML to the absorption of evanescent waves in waveguides," IEEE Microwave Wirel. Compon. Lett. 12, 218-220 (2002). [CrossRef]
- Y. Rickard and N. Georgieva, "Problem-independent enhancement of PML ABC for the FDTD method," IEEE Trans. Antennas Propag. 51, 3002-3006 (2003). [CrossRef]
- S. Cummer, "Perfectly matched layer behavior in negative refractive index materials," IEEE Antennas Wirel. Propag. Lett. 3, 172-175 (2004). [CrossRef]
- A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975).
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